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Theorem toponcom 22829
Description: If 𝐾 is a topology on the base set of topology 𝐽, then 𝐽 is a topology on the base of 𝐾. (Contributed by Mario Carneiro, 22-Aug-2015.)
Assertion
Ref Expression
toponcom ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐽)) β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐾))

Proof of Theorem toponcom
StepHypRef Expression
1 toponuni 22815 . . . 4 (𝐾 ∈ (TopOnβ€˜βˆͺ 𝐽) β†’ βˆͺ 𝐽 = βˆͺ 𝐾)
21eqcomd 2734 . . 3 (𝐾 ∈ (TopOnβ€˜βˆͺ 𝐽) β†’ βˆͺ 𝐾 = βˆͺ 𝐽)
32anim2i 616 . 2 ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐽)) β†’ (𝐽 ∈ Top ∧ βˆͺ 𝐾 = βˆͺ 𝐽))
4 istopon 22813 . 2 (𝐽 ∈ (TopOnβ€˜βˆͺ 𝐾) ↔ (𝐽 ∈ Top ∧ βˆͺ 𝐾 = βˆͺ 𝐽))
53, 4sylibr 233 1 ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOnβ€˜βˆͺ 𝐽)) β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐾))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆͺ cuni 4908  β€˜cfv 6548  Topctop 22794  TopOnctopon 22811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-topon 22812
This theorem is referenced by:  toponcomb  22830  kgencn3  23461
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